Presented to the MAA Special Session on Proof in Mathematics Education, January, 1999.

Proof as a Convincing Communication that Answers -- Why?

by David W. Henderson

Department of Mathematics, Cornell University, Ithaca, NY 14853-7901

I believe that much of the problem of teaching proofs and proving in the undergraduate curriculum is that we do not give a useful definition of what we mean by "proof". There is a formal definition of proof but, in my experience, this is not what most mathematicians use; and I find that the formal notion of proof deadens the class and the students learning. I propose the following definition as being closer to the way that mathematicians actually work and closer to how we want our students to work.

A proof is a convincing communication that answer -- Why?

It is not formal proofs -- computers can now find and check these. What we need are alive human proofs which:

are communications -- when we prove something we are not done until we can communicate it to others and the nature of this communication, of course, depends on the community to which one is communicating and is thus in part a social phenomenon.

are convincing -- a proof "works" when it convinces others. Of course some persons become convinced too easily so we are more confident in the proof if it convinces some one who was originally a skeptic. Also, a proof that convinces me may not convince you or my students.

answer -- Why? -- The proof should explain, especially it should explain something that the hearer of the proof wants to have explained. I think most people in mathematics have had the experience of logically following a proof step by step but are still dissatisfied because it did not answer questions were of the sort: "Why is it true?" "Where did it come from?" "How did you see it?" "What does it mean?".

I will report on how this definition of proof has changed my teaching and changed the ways in which my students learn and desire to do proofs.

I will give examples of standard "proofs" in textbooks that deaden learning.

I will give examples on how this definition has changed my teaching.

I will give examples of student proofs stimulated by this definition.

Related References:

Atiyah, Michael, et al, Responses to "Theoretical Mathematics: Toward a cultural synthesis of mathematics and theoretical physics", by A. Jaffe and F. Quinn, Bull. Amer. Math. Soc. 30, 178-207.

Babai, L., Transparent Proofs, Focus: The Newsletter of the Mathematical Association of America, 12, n3, 1-3.

Dieudonné, Jean, Should We Teach Modern Mathematics, American Scientist 77, 16-19.

M. Fellows, A.H. Koblitz, N. Koblitz, Cultural Aspects of Mathematics Education Reform, Notices of the A.M.S., 41, 5-9.

Henderson, David, Three papers: Mathematics and Liberation, Sue is a mathematician, and Mathematics as imagination, For the Learning of Mathematics, vol. 1, No. 3, 1981.

Henderson, David, and David Pimm, Geometric Proofs and Knowledge Without Axioms, Proceedings, 1995 Annual Meeting of the Canadian Mathematics Education Study Group, Halifax, NS: Mount Saint Vincent University Press, 1995.

Henderson, David, Experiencing Geometry on Plane and Sphere, Upper Saddle River, NJ: Prentice-Hall, 1996.

Henderson, David, I learn mathematics from my students -- multiculturalism in action, For the Learning of Mathematics, 16, 34-40, 1996.

Henderson, David, Alive Mathematical Reasoning, Proceedings, 1996 Annual Meeting of the Canadian Mathematics Education Study Group, Halifax, NS: Mount Saint Vincent University Press, 27-33, 1996.

Henderson, David, Differential Geometry: A Geometric Introduction, Upper Saddle River, NJ: Prentice-Hall, 1998.

Hersh, Reuben, What is Mathematics, Really?, New York: Oxford University Press, 1997.

Hilbert, David and S. Cohn-Vossen, Geometry and the Imagination, New York: Chelsea Publishing Co., translation copyright 1983.

Lo, Jane-Jane, Kelly Gaddis, David Henderson, Building Upon Student Experiences in a College Geometry Course, For the Learning of Mathematics, 16, no. 2, 1996.

Thurston, William P., On Proof and Progress in Mathematics, Bull. Amer. Math. Soc. 30, 161-177.